Call the three terms $ a $, $ a+d $, and $ a+2d $, in increasing order. We are told that the largest and smallest terms differ by 14, so $ (a+2d)-a=2d=14 $ or $ d=7 $. After adding half the smallest term to each term, they become $ \frac32a $, $ \frac32a+7 $, and $ \frac32a+14 $. Their sum is $ \frac92a+21=120 $, which gives $ \frac92a=99 $ or $ a=\boxed{22} $.